3.2775 \(\int (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{5/2} \, dx\)

Optimal. Leaf size=280 \[ -\frac{23763809947 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{6219281250 \sqrt{33}}+\frac{2}{75} (1-2 x)^{5/2} (3 x+2)^{3/2} (5 x+3)^{7/2}+\frac{106 (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{7/2}}{4875}+\frac{8038 \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}}{804375}+\frac{364267 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{7/2}}{36196875}-\frac{26534891 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{760134375}-\frac{359748241 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{1520268750}-\frac{23763809947 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{13682418750}-\frac{1580201444291 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{12438562500 \sqrt{33}} \]

[Out]

(-23763809947*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/13682418750 - (359748241*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*
(3 + 5*x)^(3/2))/1520268750 - (26534891*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/760134375 + (364267*Sqrt[
1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(7/2))/36196875 + (8038*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)^(7/2))/804375
+ (106*(1 - 2*x)^(3/2)*(2 + 3*x)^(3/2)*(3 + 5*x)^(7/2))/4875 + (2*(1 - 2*x)^(5/2)*(2 + 3*x)^(3/2)*(3 + 5*x)^(
7/2))/75 - (1580201444291*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(12438562500*Sqrt[33]) - (2376380
9947*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(6219281250*Sqrt[33])

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Rubi [A]  time = 0.123476, antiderivative size = 280, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {101, 154, 158, 113, 119} \[ \frac{2}{75} (1-2 x)^{5/2} (3 x+2)^{3/2} (5 x+3)^{7/2}+\frac{106 (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{7/2}}{4875}+\frac{8038 \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}}{804375}+\frac{364267 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{7/2}}{36196875}-\frac{26534891 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{760134375}-\frac{359748241 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{1520268750}-\frac{23763809947 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{13682418750}-\frac{23763809947 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{6219281250 \sqrt{33}}-\frac{1580201444291 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{12438562500 \sqrt{33}} \]

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^(5/2)*(2 + 3*x)^(3/2)*(3 + 5*x)^(5/2),x]

[Out]

(-23763809947*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/13682418750 - (359748241*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*
(3 + 5*x)^(3/2))/1520268750 - (26534891*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/760134375 + (364267*Sqrt[
1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(7/2))/36196875 + (8038*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)^(7/2))/804375
+ (106*(1 - 2*x)^(3/2)*(2 + 3*x)^(3/2)*(3 + 5*x)^(7/2))/4875 + (2*(1 - 2*x)^(5/2)*(2 + 3*x)^(3/2)*(3 + 5*x)^(
7/2))/75 - (1580201444291*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(12438562500*Sqrt[33]) - (2376380
9947*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(6219281250*Sqrt[33])

Rule 101

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a +
b*x)^m*(c + d*x)^n*(e + f*x)^(p + 1))/(f*(m + n + p + 1)), x] - Dist[1/(f*(m + n + p + 1)), Int[(a + b*x)^(m -
1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[c*m*(b*e - a*f) + a*n*(d*e - c*f) + (d*m*(b*e - a*f) + b*n*(d*e - c*f))
*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[m, 0] && GtQ[n, 0] && NeQ[m + n + p + 1, 0] && (Integ
ersQ[2*m, 2*n, 2*p] || (IntegersQ[m, n + p] || IntegersQ[p, m + n]))

Rule 154

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(h*(a + b*x)^m*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 2)), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2
*n, 2*p]

Rule 158

Int[((g_.) + (h_.)*(x_))/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol]
:> Dist[h/f, Int[Sqrt[e + f*x]/(Sqrt[a + b*x]*Sqrt[c + d*x]), x], x] + Dist[(f*g - e*h)/f, Int[1/(Sqrt[a + b*
x]*Sqrt[c + d*x]*Sqrt[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && SimplerQ[a + b*x, e + f*x] &&
SimplerQ[c + d*x, e + f*x]

Rule 113

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[(2*Rt[-((b*e
- a*f)/d), 2]*EllipticE[ArcSin[Sqrt[a + b*x]/Rt[-((b*c - a*d)/d), 2]], (f*(b*c - a*d))/(d*(b*e - a*f))])/b, x
] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &&  !LtQ[-((b*c - a*d)/d),
0] &&  !(SimplerQ[c + d*x, a + b*x] && GtQ[-(d/(b*c - a*d)), 0] && GtQ[d/(d*e - c*f), 0] &&  !LtQ[(b*c - a*d)
/b, 0])

Rule 119

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[(2*Rt[-(b/d
), 2]*EllipticF[ArcSin[Sqrt[a + b*x]/(Rt[-(b/d), 2]*Sqrt[(b*c - a*d)/b])], (f*(b*c - a*d))/(d*(b*e - a*f))])/(
b*Sqrt[(b*e - a*f)/b]), x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[(b*c - a*d)/b, 0] && GtQ[(b*e - a*f)/b, 0] &
& PosQ[-(b/d)] &&  !(SimplerQ[c + d*x, a + b*x] && GtQ[(d*e - c*f)/d, 0] && GtQ[-(d/b), 0]) &&  !(SimplerQ[c +
d*x, a + b*x] && GtQ[(-(b*e) + a*f)/f, 0] && GtQ[-(f/b), 0]) &&  !(SimplerQ[e + f*x, a + b*x] && GtQ[(-(d*e)
+ c*f)/f, 0] && GtQ[(-(b*e) + a*f)/f, 0] && (PosQ[-(f/d)] || PosQ[-(f/b)]))

Rubi steps

\begin{align*} \int (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{5/2} \, dx &=\frac{2}{75} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{7/2}-\frac{2}{75} \int \left (-\frac{113}{2}-\frac{159 x}{2}\right ) (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{5/2} \, dx\\ &=\frac{106 (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{7/2}}{4875}+\frac{2}{75} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{7/2}-\frac{4 \int \left (-3084-\frac{12057 x}{4}\right ) \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2} \, dx}{14625}\\ &=\frac{8038 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{804375}+\frac{106 (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{7/2}}{4875}+\frac{2}{75} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{7/2}-\frac{8 \int \frac{\sqrt{2+3 x} (3+5 x)^{5/2} \left (-\frac{1010595}{8}+\frac{1092801 x}{8}\right )}{\sqrt{1-2 x}} \, dx}{2413125}\\ &=\frac{364267 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{7/2}}{36196875}+\frac{8038 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{804375}+\frac{106 (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{7/2}}{4875}+\frac{2}{75} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{7/2}+\frac{8 \int \frac{(3+5 x)^{5/2} \left (\frac{108689433}{16}+\frac{79604673 x}{8}\right )}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{108590625}\\ &=-\frac{26534891 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{760134375}+\frac{364267 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{7/2}}{36196875}+\frac{8038 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{804375}+\frac{106 (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{7/2}}{4875}+\frac{2}{75} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{7/2}-\frac{8 \int \frac{\left (-\frac{5294426955}{8}-\frac{16188670845 x}{16}\right ) (3+5 x)^{3/2}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{2280403125}\\ &=-\frac{359748241 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{1520268750}-\frac{26534891 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{760134375}+\frac{364267 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{7/2}}{36196875}+\frac{8038 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{804375}+\frac{106 (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{7/2}}{4875}+\frac{2}{75} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{7/2}+\frac{8 \int \frac{\sqrt{3+5 x} \left (\frac{1390090964715}{32}+\frac{1069371447615 x}{16}\right )}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{34206046875}\\ &=-\frac{23763809947 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{13682418750}-\frac{359748241 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{1520268750}-\frac{26534891 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{760134375}+\frac{364267 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{7/2}}{36196875}+\frac{8038 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{804375}+\frac{106 (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{7/2}}{4875}+\frac{2}{75} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{7/2}-\frac{8 \int \frac{-\frac{22509028090305}{16}-\frac{71109064993095 x}{32}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{307854421875}\\ &=-\frac{23763809947 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{13682418750}-\frac{359748241 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{1520268750}-\frac{26534891 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{760134375}+\frac{364267 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{7/2}}{36196875}+\frac{8038 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{804375}+\frac{106 (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{7/2}}{4875}+\frac{2}{75} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{7/2}+\frac{23763809947 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{12438562500}+\frac{1580201444291 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{136824187500}\\ &=-\frac{23763809947 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{13682418750}-\frac{359748241 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{1520268750}-\frac{26534891 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{760134375}+\frac{364267 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{7/2}}{36196875}+\frac{8038 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{804375}+\frac{106 (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{7/2}}{4875}+\frac{2}{75} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{7/2}-\frac{1580201444291 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{12438562500 \sqrt{33}}-\frac{23763809947 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{6219281250 \sqrt{33}}\\ \end{align*}

Mathematica [A]  time = 0.239505, size = 119, normalized size = 0.42 \[ \frac{\sqrt{2} \left (1580201444291 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-795995716040 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )\right )+30 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (547296750000 x^6+579573225000 x^5-352885207500 x^4-487924998750 x^3+59959633500 x^2+157612390605 x+9093216326\right )}{410472562500} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x)^(5/2)*(2 + 3*x)^(3/2)*(3 + 5*x)^(5/2),x]

[Out]

(30*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(9093216326 + 157612390605*x + 59959633500*x^2 - 487924998750*x^
3 - 352885207500*x^4 + 579573225000*x^5 + 547296750000*x^6) + Sqrt[2]*(1580201444291*EllipticE[ArcSin[Sqrt[2/1
1]*Sqrt[3 + 5*x]], -33/2] - 795995716040*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/410472562500

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Maple [C]  time = 0.01, size = 170, normalized size = 0.6 \begin{align*}{\frac{1}{12314176875000\,{x}^{3}[phone]\,{x}^{2}-2873307937500\,x-2462835375000}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 492567075000000\,{x}^{9}[phone]\,{x}^{8}-32623479000000\,{x}^{7}-902847084300000\,{x}^{6}[phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) -1580201444291\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) -312921865912500\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}-37617016792110\,{x}^{2}-30279805737360\,x-1636778938680 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^(5/2)*(2+3*x)^(3/2)*(3+5*x)^(5/2),x)

[Out]

1/410472562500*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(492567075000000*x^9+899250660000000*x^8-326234790000
00*x^7-902847084300000*x^6+795995716040*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*EllipticF(1/11*(66+1
10*x)^(1/2),1/2*I*66^(1/2))-1580201444291*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*EllipticE(1/11*(66
+110*x)^(1/2),1/2*I*66^(1/2))-312921865912500*x^5+349206885747000*x^4+192171420950850*x^3-37617016792110*x^2-3
0279805737360*x-1636778938680)/(30*x^3+23*x^2-7*x-6)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\right )}^{\frac{3}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(2+3*x)^(3/2)*(3+5*x)^(5/2),x, algorithm="maxima")

[Out]

integrate((5*x + 3)^(5/2)*(3*x + 2)^(3/2)*(-2*x + 1)^(5/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (300 \, x^{5} + 260 \, x^{4} - 137 \, x^{3} - 136 \, x^{2} + 15 \, x + 18\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(2+3*x)^(3/2)*(3+5*x)^(5/2),x, algorithm="fricas")

[Out]

integral((300*x^5 + 260*x^4 - 137*x^3 - 136*x^2 + 15*x + 18)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)**(5/2)*(2+3*x)**(3/2)*(3+5*x)**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\right )}^{\frac{3}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(2+3*x)^(3/2)*(3+5*x)^(5/2),x, algorithm="giac")

[Out]

integrate((5*x + 3)^(5/2)*(3*x + 2)^(3/2)*(-2*x + 1)^(5/2), x)